Question

ii please

b) For the following DE, do the following: a) Find the solution in phase - amplitude form. If we think of each equation as describing a linear mass-spring system, determine how often the mass crosses the equilibrium position. Find the time at which the mass first crosses the equilibrium position. Estimate the time for which «(1< 1/100 i) "+4x'+8x = 0; x(0) = 1; x'(0) = 2 x"+2x'+10 x = 0; x(0) = 1; x'(0) = 1 4x"+4 x'+ 13 x = 0; x(0) = -1; x'0) = -1 -x"+ - x'+ - x = 0; x(0) = 2; x'(0) = -1 2 2 2

Answer 1

Given D e is Given conditions are x 10) = 1 and 10) = 1 2 Consider tax't loro +2D +10) <=0 fenin se The auxiliary on in tim) ao + 2m +10 cor > * compare with amtebm+c=0 6² = 4, yara 4.1.20= 40 6² <4 aC a The springe system is under-damped m = -2 + √2) ² - 4.1.10 2.1 - 2 I 4.40 =-23 -36 The roots are complex conjugates. i xect) = ē (a casata y sin st] Given that xo) = 1 xco) = è (c. Coscolt (a sin(o] 1 = [G.Itc,.] (CED x'lt) = - et [c, cosst et C, sin st] te* [-3c, sinst +322 cosst] &'(o)=-1(9.146.0] +1 (-30, (0) + 362 (1)]

ventication tion . ' = -, +342 1 = -1 + 362 30 = 2 = C : , :: El and C = ... 5464) = ēt [ cos (34) + s sincses] : : x (o)= el cossos + 2 / 3 sinco)] =i[ite] El .:X60) = 1) alt) = - et leoszt + 2 sin st] + et (-asinst + acosst] aco) = -4 [1+.0) ++ (-36) 42013] a-1 + 2 - veritied Hence the solution of the ginen sharing sletem DE is and) = ê( Coszt + sin (st)] O. as t alth 2,